Herz-Schur Multipliers Research Articles

Curvature-Dimension Conditions for Symmetric Quantum Markov Semigroups

Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet–Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz–Schur multipliers over group algebras and generalized depolarizing semigroups.

Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an L2-form of Hörmander's kernel condition suffices for the weak type (1,1) of Calderón-Zygmund operators acting on matrix-valued functions. To that end, we introduce an improved CZ decomposition for martingale filtrations in von Neumann algebras, and apply a very simple unconventional argument which notably avoids pseudolocalization. In second place, we establish as well the weak L1 endpoint for matrix-valued CZ operators over nondoubling measures of polynomial growth, in the line of the work of Tolsa and Nazarov/Treil/Volberg. The above results are valid for other von Neumann algebras and solve in the positive two open problems formulated in 2009. An even more interesting problem is the lack of L1 endpoint inequalities for singular Fourier and Schur multipliers over nonabelian groups. Given a locally compact group G equipped with a conditionally negative length ψ:G→R+, we prove that Herz-Schur multipliers with symbol m∘ψ satisfying a Mikhlin condition in terms of the ψ-cocycle dimension are of weak type (1,1). Our result extends to Fourier multipliers for amenable groups and imposes sharp regularity conditions on the symbol. The proof crucially combines our new CZ methods with novel forms of recent transference techniques. This L1 endpoint gives a very much expected inequality which complements the L∞→BMO estimates proved in 2014 by Junge, Mei and Parcet.

Completely Compact Herz–Schur Multipliers of Dynamical Systems

We prove that if G is a discrete group and (A,G,alpha ) is a C*-dynamical system such that the reduced crossed product Artimes _{r,alpha } G possesses property (SOAP) then every completely compact Herz–Schur (A,G,alpha )-multiplier can be approximated in the completely bounded norm by Herz–Schur (A,G,alpha )-multipliers of finite rank. As a consequence, if G has the approximation property (AP) then the completely compact Herz–Schur multipliers of A(G) coincide with the closure of A(G) in the completely bounded multiplier norm. We study the class of invariant completely compact Herz–Schur multipliers of Artimes _{r,alpha } G and provide a description of this class in the case of the irrational rotation algebra.

Multipliers and Duality for Group Actions

We define operator-valued Schur and Herz–Schur multipliers in terms of module actions, and show that the standard properties of these multipliers follow from well-known facts about these module actions and duality theory for group actions. These results are applied to study the Herz–Schur multipliers of an abelian group acting on its Pontryagin dual: it is shown that a natural subset of these Herz–Schur multipliers can be identified with the classical Herz–Schur multipliers of the direct product of the group with its dual group.

Amenable dynamical systems over locally compact groups

AbstractWe establish several new characterizations of amenable $W^*$ - and $C^*$ -dynamical systems over arbitrary locally compact groups. In the $W^*$ -setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz–Schur multipliers of $(M,G,\alpha )$ converging point weak* to the identity of $G\bar {\ltimes }M$ . In the $C^*$ -setting, we prove that amenability of $(A,G,\alpha )$ is equivalent to an analogous Herz–Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$ , as well as a particular case of the positive weak approximation property of Bédos and Conti [On discrete twisted $C^*$ -dynamical systems, Hilbert $C^*$ -modules and regularity. Münster J. Math.5 (2012), 183–208] (generalized to the locally compact setting). When $Z(A^{**})=Z(A)^{**}$ , it follows that amenability is equivalent to the 1-positive approximation property of Exel and Ng [Approximation property of $C^*$ -algebraic bundles. Math. Proc. Cambridge Philos. Soc.132(3) (2002), 509–522]. In particular, when $A=C_0(X)$ is commutative, amenability of $(C_0(X),G,\alpha )$ coincides with topological amenability of the G-space $(G,X)$ .

Fixed points and limits of convolution powers of contractive quantum measures

We study fixed points of contractive convolution operators associated to contractive quantum measures on locally compact quantum groups. We characterise the existence of non-zero fixed points respectively on $L^\infty(\mathbb{G})$ and on $C_0(\mathbb{G})$, and exploit these results to obtain for example the structure of the fixed points on the non-commutative $L_p$-spaces. Some consequences for the fixed points of classical convolution operators and Herz-Schur multipliers are also indicated.

Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full <I>C</I><SUP>*</SUP>-Algebras

In this paper we develop the notion of Schur multipliers and Herz-Schur multipliers to the context of Fell bundle, as a generalization of the theory of multipliers of locally compact groups and crossed products. We prove a characterization theorem of this generalized Schur multiplier in terms of the representation of Fell bundles. In order to prove this characterization theorem we define a new class of completely bounded maps; and discuss in detail of its properties. In this process, by the way, we give a new proof of Stinpring’s Theorem of non-unital version. Then we investigate the transference theorem of Schur multipliers and Herz-Schur multipliers, which is a generalization of the transference theorem well-known either in the group case or crossed products. We use the notion of multipliers to define an approximation property of Fell bundles. Then we give a necessary and sufficient condition if the reduced cross-sectional algebra of a Fell bundle over a discrete groups is nuclear in terms of this generalized notion. This is a generalization of the classical theorem concerning the amenability of locally compact groups. As an application, we prove that for a Fell bundle, if its cross-sectional algebra is nuclear, then for any subgroup of the group on which the Fell bundle is defined, the cross-sectional algebra of the restricted Fell bundle on this subgroup is nuclear.

Weak amenability for dynamical systems

Using the recently developed notion of a Herz-Schur multiplier of a C*-dynamical system we introduce weak amenability of C*- and W*-dynamical systems. As a special case we recover Haagerup's characterisation of weak amenability of a discrete group. We also consider a generalisation of the Fourier algebra and its multipliers to crossed products.

Exactness and SOAP of crossed products via Herz–Schur multipliers

Given a C⁎-dynamical system (A,G,α), with G a discrete group, Schur A-multipliers and Herz–Schur (A,G,α)-multipliers are used to implement approximation properties, namely exactness and the strong operator approximation property (SOAP), of A⋊α,rG. The resulting characterisations of exactness and SOAP of A⋊α,rG generalise the corresponding statements for the reduced group C⁎-algebra.

Spherical-Radial Multipliers on the Heisenberg Group

Let Hn be the (2n+1)-dimensional Heisenberg group. We consider a radial Fourier multiplier which is a spherical function on Hn and show that it is a Herz-Schur multiplier.

Herz–Schur multipliers of dynamical systems

We extend the notion of a Herz–Schur multiplier to the setting of non-commutative dynamical systems: given a C*-algebra A, a locally compact group G, and an action α of G on A, we define transformations on the reduced crossed product of A by α which, in the case A=C, reduce to the classical Herz–Schur multipliers. We introduce Schur A-multipliers, establish a characterisation that generalises the classical descriptions of Schur multipliers, and present a transference theorem in the new setting, identifying isometrically the Herz–Schur multipliers of the dynamical system (A,G,α) with the invariant part of the Schur A-multipliers. We discuss special classes of Herz–Schur multipliers, in particular, those associated to a locally compact abelian group G and its canonical action on the C*-algebra C⁎(Γ) of the dual group Γ.

Weak Haagerup property of dynamical systems

ABSTRACTUsing Herz-Schur multipliers of a -dynamical system, we introduce the weak Haagerup property of a -dynamical system and prove that it is equivalent to the weak Haagerup property of the associated -crossed product. On the other hand, we investigate an additional assumption under which the weak Haagerup property of the original -algebra and the group implies that of the associated -dynamical system. We also define and study the weak Haagerup property of a -dynamical system.

Positive Herz–Schur multipliers and approximation properties of crossed products

For aC*-algebraAand a setXwe give a Stinespring-type characterisation of the completely positive SchurA-multipliers on κ(ℓ2(X)) ⊗A. We then relate them to completely positive Herz–Schur multipliers onC*-algebraic crossed products of the formA⋊α,rG, withGa discrete group, whose various versions were considered earlier by Anantharaman-Delaroche, Bédos and Conti, and Dong and Ruan. The latter maps are shown to implement approximation properties, such as nuclearity or the Haagerup property, forA⋊α,rG.

Ideals of the Fourier algebra, supports and harmonic operators

AbstractWe examine the common null spaces of families of Herz–Schur multipliers and apply our results to study jointly harmonic operators and their relation with jointly harmonic functionals. We show how an annihilation formula obtained in [1] can be used to give a short proof as well as a generalisation of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite function. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other.

Semigroups of Herz–Schur multipliers

In order to investigate the relationship between weak amenability and the Haagerup property for groups, we introduce the weak Haagerup property, and we prove that having this approximation property is equivalent to the existence of a semigroup of Herz–Schur multipliers generated by a proper function (see Theorem 1.2). It is then shown that a (not necessarily proper) generator of a semigroup of Herz–Schur multipliers splits into a positive definite kernel and a conditionally negative definite kernel. We also show that the generator has a particularly pleasant form if and only if the group is amenable. In the second half of the paper we study semigroups of radial Herz–Schur multipliers on free groups. We prove that a generator of such a semigroup is linearly bounded by the word length function (see Theorem 1.6).

Radial multipliers on reduced free products of operator algebras

Let Ai be a family of unital C⁎-algebras, respectively, of von Neumann algebras and ϕ:N0→C. We show that if a Hankel matrix related to ϕ is trace-class, then there exists a unique completely bounded map Mϕ on the reduced free product of the Ai, which acts as a radial multiplier. Hereby we generalize a result of Wysoczański for Herz–Schur multipliers on reduced group C⁎-algebras for free products of groups.

Completely co-bounded Schur multipliers

A linear map u: E → F between operator spaces is called completely co-bounded if it is completely bounded as a map from E to the opposite of F. We give several simple results about completely co-bounded Schur multipliers on B(2) and the Schatten class Sp .W e also consider Herz-Schur multipliers on groups.

The Herz–Schur multiplier norm of sets satisfying the Leinert condition

It is well known that in a free group $\def\F{{\mathbb F}}\F$, one has $\def\F{{\mathbb F}}\|\chi_E\|_{M_{cb}A(\F)}\leq 2$, where $E$ is the set of all the generators. We show that the (completely) bounded multiplier norm of any set satisfying the Leinert

Herz-Schur multipliers and weakly almost periodic functions on locally compact groups

For a locally compact group G G and 1 > p > ∞ 1>p>\infty , let A p ( G ) A_{p}(G) be the Herz–Figà-Talamanca algebra and B p ( G ) B_{p}(G) the Herz-Schur multipliers of G G , and M A p ( G ) MA_{p}(G) the multipliers of A p ( G ) A_{p}(G) . Let W ( G ) W(G) be the algebra of continuous weakly almost periodic functions on G G . In this paper, we show that (1), if G G is a noncompact nilpotent group or a noncompact [IN]-group, then W ( G ) / B p ( G ) − W(G)/B_{p}(G)^{-} contains a linear isometric copy of l ∞ ( N ) l^{\infty }({\mathbb {N}}) ; (2), for a noncommutative free group F , B p ( F ) F, B_{p}(F) is a proper subset of M A p ( F ) ∩ W ( F ) {MA_{p}(F)\cap {W(F)}} .

A Characterization of Radial Herz-Schur Multipliers on Free Products of Discrete Groups

A characterization of radial Herz-Schur multipliers on groups which are free products G = ∗ N i=1 G i of subgroups of the same order (finite or infinite) is given. The multiplier norm is computed in terms of trace norms of appropriate trace operators. This applies in particular to free groups F N = ∗ N i=1 Z with block length. This method also applies to characterizing radial multipliers of free product groups of distinct orders, all greater than 2, but in this case we can compute only the multiplier norm up to some constant.